Found problems: 85335
Two cars travel along a circular track $n$ miles long, starting at the same point. One car travels $25$ miles along the track in some direction. The other car travels $3$ miles along the track in some direction. Then the two cars are once again at the same point along the track. If $n$ is a positive integer, find the sum of all possible values of $n.$
A point $ (x,y)$ in the plane is called a lattice point if both $ x$ and $ y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to
$ \textbf{(A)}\ 4.0\qquad
\textbf{(B)}\ 4.2\qquad
\textbf{(C)}\ 4.5\qquad
\textbf{(D)}\ 5.0\qquad
\textbf{(E)}\ 5.6$
Determine
$$\lim_{x\to\infty}\frac{\sqrt{x+2014}}{\sqrt x+\sqrt{x+2014}}$$
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that all real numbers $x$ and $y$ satisfy \[f(f(x)+y)=f(x^2-y)+4f(x)y.\]
Let $ABC$ a triangle and $P$ a point such that the triangles $PAB, PBC, PCA$ have the same area and the same perimeter. Prove that if:
a) $P$ is in the interior of the triangle $ABC$ then $ABC$ is equilateral.
b) $P$ is in the exterior of the triangle $ABC$ then $ABC$ is right angled triangle.
In a circle $\Gamma_{1}$, centered at $O$, $AB$ and $CD$ are two unequal in length chords intersecting at $E$ inside $\Gamma_{1}$. A circle $\Gamma_{2}$, centered at $I$ is tangent to $\Gamma_{1}$ internally at $F$, and also tangent to $AB$ at $G$ and $CD$ at $H$. A line $l$ through $O$ intersects $AB$ and $CD$ at $P$ and $Q$ respectively such that $EP = EQ$. The line $EF$ intersects $l$ at $M$. Prove that the line through $M$ parallel to $AB$ is tangent to $\Gamma_{1}$
Find all positive integers $n, n>1$ for wich holds :
If $a_1, a_2 ,\dots ,a_k$ are all numbers less than $n$ and relatively prime to $n$ , and holds $a_1<a_2<\dots <a_k $, then none of sums $a_i+a_{i+1}$ for $i=1,2,3,\dots k-1 $ are divisible by $3$.
Find all real numbers $a$, $b$ and $c$ that satisfy the following system of equations:
$$\begin{cases}
ab-c = 3 \\
a+bc = 4 \\
a^2+c^2 = 5\end{cases}$$
Determine the smallest positive integer $k$ so that the equation $$2002x+273y=200201+k$$ has integer solutions, and for that value of $k$, find the number of solutions $\left (x,y\right )$ with $x$, $y$ positive integers that have the equation.
Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$
Show that there do not exist natural numbers $a_1, a_2, \dots, a_{2018}$ such that the numbers \[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \] are all powers of $5$
[i]Proposed by Tejaswi Navilarekallu[/i]
A gambler has $\$25$ and each turn, if the gambler has a positive amount of money, a fair coin is flipped. If it is heads, the gambler gains a dollar and if it is tails, the gambler loses a dollar. But, if the gambler has no money, he will automatically be given a dollar (which counts as a turn). What is the expected number of turns for the gambler to double his money?
Source: 1976 Euclid Part A Problem 4
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The points $(1,y_1)$ and $(-1,y_2)$ lie on the curve $y=px^2+qx+5$. If $y_1+y_2=14$, then the value of $p$ is
$\textbf{(A) } 2 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 2-q \qquad \textbf{(E) }\text{none of these}$
A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to $(1,0)$. Then it makes a $90^\circ$ counterclockwise and travels $\frac 12$ a unit to $\left(1, \frac 12 \right)$. If it continues in this fashion, each time making a $90^\circ$ degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest?
$\text{(A)} \ \left(\frac 23, \frac 23 \right) \qquad \text{(B)} \ \left( \frac 45, \frac 25 \right) \qquad \text{(C)} \ \left( \frac 23, \frac 45 \right) \qquad \text{(D)} \ \left(\frac 23, \frac 13 \right) \qquad \text{(E)} \ \left(\frac 25, \frac 45 \right)$
Let $n$ be a positive integer. Suppose for any $\mathcal{S} \subseteq \{1, 2, \cdots, n\}$, $f(\mathcal{S})$ is the set containing all positive integers at most $n$ that have an odd number of factors in $\mathcal{S}$. How many subsets of $\{1, 2, \cdots, n\}$ can be turned into $\{1\}$ after finitely many (possibly zero) applications of $f$?
Find all integer solutions to the equation $x^2=y^2(x+y^4+2y^2)$ .
There are a board with $2n \cdot 2n \ (= 4n^2)$ squares and $4n^2-1$ cards numbered with different natural numbers. These cards are put one by one on each of the squares. One square is empty. We can move a card to an empty square from one of the adjacent squares (two squares are adjacent if they have a common edge). Is it possible to exchange two cards on two adjacent squares of a column (or a row) in a finite number of movements?
How many ways are there to place the integers from $1$ to $8$ on the vertices of a regular octagon such that the sum of the numbers on any $4$ vertices forming a rectangle is even? Rotations and reflections of the same arrangement are considered distinct
Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$.
Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.
Find the number of ways to order the integers $1,2,3,4,5,6,7$ from left to right so that each integer has all its divisors besides itself appearing to the left of it.
If $x\in\left[-\frac{5\pi}{12},-\frac{\pi}{3}\right]$, then the maximum value of $y=\tan\left(x+\frac{2\pi}{3}\right)-\tan\left(x+\frac{\pi}{6}\right)+\cos\left(x+\frac{\pi}{6}\right)$ is
$\text{(A)}\frac{12}{5}\sqrt2\qquad\text{(B)}\frac{11}{6}\sqrt2\qquad\text{(C)}\frac{11}{6}\sqrt3\qquad\text{(D)}\frac{12}{5}\sqrt3$
Let $ABCD$ be a square such that $A=(0,0)$ and $B=(1,1)$. $P(\frac{2}{7},\frac{1}{4})$ is a point inside the square. An ant starts walking from $P$, touches $3$ sides of the square and comes back to the point $P$. The least possible distance traveled by the ant can be expressed as $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are integers and $a$ not divisible by any square number other than $1$. What is the value of $(a+b)$?
There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base-$b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$?
$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21$
Solve in positive integers \[ x^y + y = y^x + x \]
A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?